Basic invariants
| Dimension: | $3$ |
| Group: | $A_4\times C_2$ |
| Conductor: | \(637\)\(\medspace = 7^{2} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.31213.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4\times C_2$ |
| Parity: | even |
| Determinant: | 1.13.2t1.a.a |
| Projective image: | $A_4$ |
| Projective stem field: | Galois closure of 4.0.8281.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - 3x^{3} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 2\cdot 29 + 20\cdot 29^{2} + 11\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 22 + \left(2 a + 14\right)\cdot 29 + \left(6 a + 8\right)\cdot 29^{2} + \left(7 a + 28\right)\cdot 29^{3} + \left(15 a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 1 + \left(9 a + 4\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(12 a + 11\right)\cdot 29^{3} + \left(17 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 3 + \left(19 a + 10\right)\cdot 29 + \left(17 a + 1\right)\cdot 29^{2} + \left(16 a + 4\right)\cdot 29^{3} + \left(11 a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 15 + \left(26 a + 11\right)\cdot 29 + \left(22 a + 7\right)\cdot 29^{2} + 21 a\cdot 29^{3} + \left(13 a + 21\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 + 15\cdot 29 + 8\cdot 29^{2} + 2\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,5,4,6,2,3)$ | $0$ |
| $4$ | $6$ | $(1,3,2,6,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.